The purpose of this activity is for students to understand the difference between actual data and information predicted by the linear model. This prepares students to understand that a line of best fit, while representative of the data, is different from the actual data and to start thinking about what differences occur between the two. This is useful for when students learn about residuals in the associated Algebra 1 lesson. Comparing the actual data with the model allows students to model with mathematics (MP4).

Monitor for students who use these different strategies:

• Use the points and line on the graph to answer the questions
• Use the data table and equation to answer the questions

The purpose of this discussion is for students to understand that the actual data (in the form of points on the graph and data in the table) can differ from a linear model (in the form of the line on the graph and the equation) even when the model is a good one.

Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach, and then use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “Why do the different approaches lead to the same outcome?”
• “How is using the points on the graph and the data in the table the same? How are they different? What about the line on the graph and the equation?”

Discuss how students used the graph and table to answer the questions. Here are sample questions to promote class discussion:

• “Is the table more helpful in answering questions about the actual data or the values estimated by the linear model?” (the actual data)
• “Which is more helpful in answering questions about the values estimated by the linear model: the graph, the equation, or the table?” (The graph that displays the linear model and the equation that represents the linear model. The graph is helpful in finding how many times the model and the actual sales were equivalent. The equation is helpful in finding when there was a difference of at least 25 between actual sales and the amount of sales predicted using the linear model.)
• “When asked about estimates, how do you use the graph to answer the question?” (The line on the graph shows every estimate made using the linear model . Finding a point on the line that corresponds to specific information gives additional information about estimated values.) 
• “When asked about what was actually sold, how do you use the graph to answer the question?” (The plotted points show how many sales were actually made when the eyeglasses were  dollars.)
• “How can you tell when the linear model and the actual data are equivalent?” (The plotted points are directly on the line when the two are equivalent.)
• “How can you tell when the linear model predicts a sales amount that is more than the actual sales amount?” (If the linear model predicts sales that are more than the actual amount, then the line will be above the plotted points.)

The purpose of this activity is for students to practice interpreting a linear model and understanding the difference between the actual data and the linear model predictions. Students can use the equation and graph to complete the table, which allows them to synthesize the graph, equation, and table to understand that they represent the same information. Then students use the graph, equation, and completed table to interpret the data and predictions. Students model with mathematics when evaluating the effectiveness of the model compared with the actual data (MP4).

The purpose of this discussion is for students to interpret data and predictions using a linear model that is represented in three ways: using an equation, a graph, and a table. Discuss how students used the representations to answer the questions. Here are sample questions to promote class discussion:

• “How can the linear equation be used to complete the table with estimated amounts spent on gas?” (To use the equation to complete the table, substitute the given value for and solve for the estimated .)
• “How can you use the graph to complete the estimated amount spent on gas?” (To use the graph to complete the table, find the point on the line associated with the given -value to find the estimated -value. Be sure to use the coordinates from the line, not the plotted points on the scatter plot.)